Related papers: On some coupled local and nonlocal diffusion model…
Local-nonlocal coupling approaches provide a means to combine the computational efficiency of local models and the accuracy of nonlocal models. This paper studies the continuous and discrete formulations of three existing approaches for the…
The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of…
We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our…
Cell-cell adhesion is an inherently nonlocal phenomenon. Numerous partial differential equation models with nonlocal term have been recently presented to describe this phenomenon, yet the mathematical properties of nonlocal adhesion model…
We derive the dual variational principle (principle of minimal complementary energy) for the nonlocal nonlinear scalar diffusion problem, which may be viewed as the nonlocal version of the $p$-Laplacian operator. We establish existence and…
We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components…
Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this…
A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. In this work we extract nonlocal models from data, circumventing these challenges…
In this work, we address a parabolic problem featuring a potentially doubly nonlinear term, governed by a combination of local and nonlocal operators (see Problem P1 below). We first establish the local existence of weak energy solutions…
The nonlocal-to-local asymptotics investigation for evolutionary problems is a central topic both in the theory of PDEs and in functional analysis. More recently, it became the main core of the mathematical analysis of phase-separation…
This article establishes local strong well-posedness and global strong well-posedness close to constant equilibria of a model coupling the primitive equations of ocean and atmospheric dynamics with Hibler's viscous-plastic sea ice model. In…
In this paper, we look at a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov…
A general method for extending a non-dissipative nonlinear Schr\"odinger and Liouville-von Neumann 1-particle dynamics to an arbitrary number of particles is described. It is shown at a general level that the dynamics so obtained is…
Two-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenized flow and transport equations are solved on the macroscopic scale, while effective parameters are…
In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal…
We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which…
This note is devoted to some nonlocal, nonlinear elliptic problems with an emphasis on the computation of the solution of such problems, reducing it in particular to a fixed point argument in R. Errors estimates and numerical experiments…
We study a nonlocal Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects and degenerate mobility. The nonlocality is described by means of a symmetric singular kernel. We define a notion of weak solution adapted to…
Nonlocality is important in realistic mathematical models of physical and biological systems when local models fail to capture the essential dynamics and interactions that occur over a range of distances. This review illustrates different…
We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These type of operators are used to model anomalous diffusion and, for a special…